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Singularities and infinities from the TGD point of view

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Gary Ehlenberg sent two links to Quantamagazine articles, which are very relevant (perhaps not by accident!) for what I have been working with recently.

The first link was to a very interesting article about the the role of singularities in physics. Already in twistor Grassmann approach, singularities of the scattering amplitudes turned out to be central as data determining them. Kind of holography was in question.

I have been just working with singularities of space-time surface and have made a breakthrough in the understanding of what graviton is but also in the understanding of what the fundamental vertices (actually vertex!) of the scattering amplitudes are in the TGD framework.

In holography=generalized holomorphy view space-time surfaces are minimal surfaces with generalized holomorphic imbedding to H=M4×CP2 implying the minimal surface property.

  1. The minimal surface property fails at lower-dimensional singularities taking the role of holographic data and the trace of the second fundamental form (SFF) analogous to a acceleration associated with the 4-D Bohr orbit of the particle as 3-surface has a delta function like singularity but vanishes elsewhere.
  2. The minimal surface property expressess masslessness for both fields and particles as 3-surfaces. At the singularities masslessness property fails and singularities can be said to serve as sources which in QFTs define scattering amplitudes.
  3. The singularities are analogs of poles and cuts for the 4-D generalization of the ordinary holomorphic functions. Also for the ordinary holomorphic functions the Laplace equation as analog massless field equation and expressing analyticity fails. Complex analysis generalizes to dimension 4.
  4. The conditions at the singularity give a generalization of Newton’s F=ma! I ended up where I started more than 50 years ago!
  5. In dimension 4, and only there, there is an infinite number of exotics diff structures, which differ from ordinary ones at singularities of measure zero analogous to defects. These defects correspond naturally to the singularities. For the exotic diff structure one can say that there is no singularity. This means that complex analysis generalizes to dimension 4 and only to dimension 4.
  6. Group theoretically the trace of the SFF can be regarded as a generalization of the Higgs field, which is non-vanishing only at the vertices and this is enough. Singularities take the role of generalized particle vertices and determine the scattering amplitudes. The second fundamental form contracted with the embedding space gamma matrices and slashed between the second quantized induced spinor field and its conjugate gives the universal vertex involving only fermions (bosons are bound states of fermions in TGD). It contains both gauge and gravitational contributions to the scattering amplitudes and there is a complete symmetry between gravitational and gauge interactions. Gravitational couplings come out correctly as the radius squared of CP2 as also in the classical picture.

This generalized Higgs field characterizing singularities would dictate all scattering amplitudes! Generalized Higgs would be really the God particle! Its CP2 part gives standard model interactions and M4 part gives gravitation.

Gary Ehlenberg sent another link to a Quantamagazine article (see this), which is very relevant to what I have been working on recently. I am not going to comment on the so called alien calculus discussed in the article as a proposal to get rid of the infinities of quantum field theories. Rather, I will explain how this problem is solved in the TGD framework (see

The problem of infinities is due to the assumption that the point-like nature of fundamental objects. In superstring models this problem was at least partially solved but superstrings were not the option chosen by Nature.

  1. The basic discovery of TGD is that the generalization of complex structure is possible in dimension 4 of the space-time and corresponds to the existence of exotic diff structures (see Nature wants all that it can get and has chosen the option with the maximal structural richness.
  2. In TGD particles become 3-D surfaces whose 4-D orbits are analogs of Bohr orbits with a finite non-determinism at which the minimal surface property fails. The mathematically ill-defined path integral reduces to a finite sum and only the well-defined functional integral over 3-surfaces remains. Divergences disappear completely.
  3. Scattering amplitudes reduce to sums over contributions from the lower-D singularities of the minimal surfaces. Singularities are analogous to the poles of holomorphic functions in holography=holomorphy vision and generalized holomorphic maps define an infinite-D symmetry group analogous to holomorphic maps in string models.
  4. The trace of the second fundamental form slashed between the induced free spinor fields of M42 gives the universal vertex and contains contributions of all basic interactions including gravitation. Induced spinor fields are second quantized spinor fields of H=M4×CP2 and correlation functions for these free spinor fields determine the scattering amplitudes.

See the article What gravitons are and could one detect them in the TGD Universe?) and the chapter the chapter About the Relationships Between Electroweak and Strong Interactions and Quantum Gravity in the TGD Universe.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.


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